According to the ASPCA, cats can eat salmon. Salmon is a great source of lean protein and contains omega-3 fatty acids that help improve skin and coat health. In fact, many dry cat foods contain salmon as an ingredient because it is beneficial for felines. 
But some salmon may contain heavy metals that can be harmful to cats. If you feed your cat salmon on occasion, it's important to ensure that it's cooked properly without spices, herbs, etc. 
Also, most cats love salmon, so if you give your cat salmon too often, it might ignore eating anything else. Don't make salmon the prime source of your cat's diet.